2. Photoeffect ratio R_P

Until recently it was believed that the energy dependence of R_P for helium was well understood over the whole nonrelativistic energy region. The theoretical predictions were consistent with measurements showing that R_P increases with energy, reaching a maximum (R_P=0.05) at photon energies of $\omega=150-250$ eV, and then decreases to the constant shake-off value R_P=0.017 for energies above roughly 3 keV.

New measurements [3], performed using cold target recoil spectroscopy, have found the peak of R_P to be lower than previously assumed, by about 25 %. These new data are consistent with calculations [4] relating the photoeffect ratio to that for fast charged particle impact.

The shake-off ratio is given in nonrelativistic calculations [5,6,7,8] as

$$R_{P}=1-\frac{\sum_{B}\left\vert\int\Phi_{B}^{\ast} ({\bf r}_... ...vert\Psi_{i}({\bf r}_{1},0)\right\vert^{2}d^{3} {\rm r}_{1}} .$$ (1)

Here $\Psi_{i}({\bf r_{1}},{\bf r_{2}})$ represents the initial state wave function, $\Phi_{B}({\bf r}_{1})$ is a bound state hydrogen-like electron wave function (in the potential of charge Z), and the summation is over all bound states. The shake off formula Eq. (1) has usually been derived in the nonrelativistic dipole approximation without retardation. It reflects the fact that, in the dipole approximation, the photon can be absorbed only if nucleus takes additional momentum (note that one of the electrons in Eq. (1) is removed from near the nucleus, ${\bf r}_{2}=0$). Namely, in the dipole approximation, a two-electron system, in the absence of nuclear field, cannot absorb a photon. Most theoretical analysis uses the dipole approximation. However, the shake-off result Eq. (1) is not changed if retardation effects (i.e. higher multipole effects) are included, given that for low Z two-electron systems retardation has little effect on ground state wave function $\Psi_{i}$ and hydrogenic bound states $\Phi_{B}$. In explicit calculations, using wave functions consistent with shake-off assumptions, Kornberg and Miraglia [9] have shown that, even when retardation alters the cross sections for single and double photoionization by significant factors, the same factor enters both processes, and so the ratio R_P is unchanged.

We may likewise expect that relativistic effects do not alter the photoeffect ratio R_P, based on the shake-off argument. The shake-off process, when applied to double photoionization, can be viewed as a two-step process. In the first step one electron is suddenly removed, corresponding to the single ionization process. Here we may allow very large, even relativistic, energies and therefore relativistic effects can be important. The second step is shake-off of the other electron which is (at least for small Z) most often a slow, nonrelativistic electron. Calculating the ratio between double and single photoionization cross sections, under the assumption of the validity of the sudden approximation, the contributions of the first step (the single ionization cross section), including relativistic as well as retardation effects, cancel out completely, leaving R_P as in Eq. (1).

However, at high energies, where the shake-off mechanism was assumed to be valid, Drukarev [10,11] has called attention to an additional mechanism [12,13,14] for double photoeffect, which would first manifest itself as a linear rise of the ratio with energy, causing a roughly 10% correction to the shake-off limit for R_P at about 12 keV. This prediction awaits experimental verification. The point is that, while a photon cannot be absorbed by one free electron in the absence of the atomic nucleus (and momentum must be transfered to the atomic nucleus), a photon can be absorbed by two free electrons (though not in the dipole approximation). This leads to a mechanism for double ionization which is not available for single ionization. The analysis was performed within nonrelativistic quantum mechanics without assuming dipole approximation, therefore allowing two electrons to absorb the photon even in the absence of the atomic nucleus[15].

A pair of photoelectrons produced through this alternative mechanism share the photon energy nearly equally (and we may call this mechanism an ``equal energy sharing mechanism''). The residual ion is left with small momentum. This is different from the case when photoelectrons are produced through the shake-off mechanism, in which one electron takes almost all the photon energy and the residual ion takes almost all transfered momentum. Note that in the experimental situation where photoeffect events are distinguished from Compton events by measuring the recoil of the residual ion [16,17], the events produced through the equal energy sharing mechanism are not counted in photoeffect events, but in Compton events. These experiments count as photoeffect events only those produced in the shake-off regime (one electron takes almost all available energy while the other shakes-off), and therefore such experiments would not observe the term in R_P caused by the equal energy sharing mechanism.

Even though the shake-off ratio may therefore not be the high energy limit of R_P, it is a physically well-defined quantity which we may measure. This observable, which we may call the shake-off ratio R_P^s.o., gives us information about the initial state electron-electron correlation when one of the electrons is near the nucleus [18,19].

Likewise, if the equal energy sharing mechanism is of the estimated magnitude, it may be within the scope of today's experimental methods. Its measurement should provide information on electron-electron correlation when the electrons are close to each other. This mechanism may also be important in double-electron capture processes in heavy (highly-charged) ion-atom collisions when a single high energy photon is emitted. For low Z ions the theoretical predictions are reported to be in good agreement with experimental data [20]. However, for highly charged ions a surprisingly enhanced cross section for double electron capture is found [21].